Optimal. Leaf size=304 \[ -\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (\frac {f x^m}{e}+1\right )}{e g m^2}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {b f k n x^m \log ^2(x) (g x)^{-m}}{2 e g}+\frac {b f k n x^m \log (x) (g x)^{-m}}{e g m} \]
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Rubi [A] time = 0.31, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2455, 19, 266, 36, 29, 31, 2376, 2301, 2454, 2394, 2315, 16} \[ \frac {b f k n x^m (g x)^{-m} \text {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{e g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {b f k n x^m \log ^2(x) (g x)^{-m}}{2 e g}+\frac {b f k n x^m \log (x) (g x)^{-m}}{e g m} \]
Antiderivative was successfully verified.
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Rule 16
Rule 19
Rule 29
Rule 31
Rule 36
Rule 266
Rule 2301
Rule 2315
Rule 2376
Rule 2394
Rule 2454
Rule 2455
Rubi steps
\begin {align*} \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-(b n) \int \left (\frac {f k x^{-1+m} (g x)^{-m} \log (x)}{e g}-\frac {f k x^{-1+m} (g x)^{-m} \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m x}\right ) \, dx\\ &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-\frac {(b f k n) \int x^{-1+m} (g x)^{-m} \log (x) \, dx}{e g}+\frac {(b n) \int \frac {(g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{g m}+\frac {(b f k n) \int x^{-1+m} (g x)^{-m} \log \left (e+f x^m\right ) \, dx}{e g m}\\ &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {(b n) \int (g x)^{-1-m} \log \left (d \left (e+f x^m\right )^k\right ) \, dx}{m}-\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {\log (x)}{x} \, dx}{e g}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {\log \left (e+f x^m\right )}{x} \, dx}{e g m}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {(b f k n) \int \frac {x^{-1+m} (g x)^{-m}}{e+f x^m} \, dx}{g m}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \operatorname {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^m\right )}{e g m^2}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-\frac {\left (b f^2 k n x^m (g x)^{-m}\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^m\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {1}{x \left (e+f x^m\right )} \, dx}{g m}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \operatorname {Subst}\left (\int \frac {1}{x (e+f x)} \, dx,x,x^m\right )}{g m^2}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^m\right )}{e g m^2}-\frac {\left (b f^2 k n x^m (g x)^{-m}\right ) \operatorname {Subst}\left (\int \frac {1}{e+f x} \, dx,x,x^m\right )}{e g m^2}\\ &=\frac {b f k n x^m (g x)^{-m} \log (x)}{e g m}-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 162, normalized size = 0.53 \[ \frac {(g x)^{-m} \left (-2 \left (a m+b m \log \left (c x^n\right )+b n\right ) \left (e \log \left (d \left (e+f x^m\right )^k\right )+f k x^m \log \left (f-f x^{-m}\right )\right )+2 f k m x^m \log (x) \left (a m+b m \log \left (c x^n\right )-b n \log \left (\frac {f x^m}{e}+1\right )+b n \log \left (f-f x^{-m}\right )+b n\right )-2 b f k n x^m \text {Li}_2\left (-\frac {f x^m}{e}\right )-b f k m^2 n x^m \log ^2(x)\right )}{2 e g m^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.93, size = 239, normalized size = 0.79 \[ -\frac {2 \, b f g^{-m - 1} k m n x^{m} \log \relax (x) \log \left (\frac {f x^{m} + e}{e}\right ) + 2 \, b f g^{-m - 1} k n x^{m} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) - {\left (b f k m^{2} n \log \relax (x)^{2} + 2 \, {\left (b f k m^{2} \log \relax (c) + a f k m^{2} + b f k m n\right )} \log \relax (x)\right )} g^{-m - 1} x^{m} + 2 \, {\left (b e m n \log \relax (d) \log \relax (x) + {\left (b e m \log \relax (c) + a e m + b e n\right )} \log \relax (d)\right )} g^{-m - 1} + 2 \, {\left ({\left (b f k m \log \relax (c) + a f k m + b f k n\right )} g^{-m - 1} x^{m} + {\left (b e k m n \log \relax (x) + b e k m \log \relax (c) + a e k m + b e k n\right )} g^{-m - 1}\right )} \log \left (f x^{m} + e\right )}{2 \, e m^{2} x^{m}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{-m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \left (g x \right )^{-m -1} \ln \left (d \left (f \,x^{m}+e \right )^{k}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b m \log \left (x^{n}\right ) + {\left (m \log \relax (c) + n\right )} b + a m\right )} g^{-m - 1} \log \left ({\left (f x^{m} + e\right )}^{k}\right )}{m^{2} x^{m}} + \int \frac {b e m \log \relax (c) \log \relax (d) + a e m \log \relax (d) + {\left ({\left (f k m + f m \log \relax (d)\right )} a + {\left (f k n + {\left (f k m + f m \log \relax (d)\right )} \log \relax (c)\right )} b\right )} x^{m} + {\left (b e m \log \relax (d) + {\left (f k m + f m \log \relax (d)\right )} b x^{m}\right )} \log \left (x^{n}\right )}{f g^{m + 1} m x x^{2 \, m} + e g^{m + 1} m x x^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (g\,x\right )}^{m+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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